Retrieval under informative initialization with finite sample complexity and finite iterations

Determine whether alternating minimization for bilinear regression with square loss and i.i.d. Gaussian covariates retrieves the target vectors under finite sample complexity κ = P/N = O(1) and a finite number of iterations when the initialization is informative, specifically when the initialization correlation satisfies 0 ≠ m0 = O(1).

Background

The authors prove that AM fails to retrieve the target under random initialization (m0 = 0) with finite iterations and finite sample complexity. Motivated by numerical evidence, they put forward a contrasting conjecture for informative initialization (nonzero correlation with the target), suggesting that retrieval may be possible without requiring diverging iterations or sample complexity.

A definitive resolution would clarify the minimal initialization and data requirements for successful recovery in this full-batch AM setting and delineate the regime where recovery occurs.